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The Project Gutenberg EBook of Solid Geometry with Problems and Applications (Revised edition), by H. E. Slaught and N. J. Lennes This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Solid Geometry with Problems and Applications (Revised edition) Author: H. E. Slaught N. J. Lennes Release Date: August 26, 2009 [EBook #29807] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK SOLID GEOMETRY *** Bonaventura Cavalieri (1598–1647) was one of the most influential mathematicians of his time. He was chiefly noted for his invention of the so-called “Principle of Indivisibles” by which he derived areas and volumes. See pages 143 and 214. SOLID GEOMETRY WITH PROBLEMS AND APPLICATIONS REVISED EDITION BY H. E. SLAUGHT, Ph.D., Sc.D. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO AND N. J. LENNES, Ph.D. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MONTANA ALLYN and BACON Bo<on New York Chicago Produced by Peter Vachuska, Andrew D. Hwang, Chuck Greif and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber’s note The original book is copyright, 1919, by H. E. Slaught and N. J. Lennes. Figures may have been moved with respect to the surrounding text. Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for printing, but may be easily recompiled for screen viewing. Please see the preamble of the L A T E X source file for instructions. PREFACE In re-writing the Solid Geometry the authors have consistently car- ried out the distinctive features described in the preface of the Plane Geometry. Mention is here made only of certain matters which are particularly emphasized in the Solid Geometry. Owing to the greater maturity of the pupils it has been possible to make the logical structure of the Solid Geometry more prominent than in the Plane Geometry. The axioms are stated and applied at the precise points where they are to be used. Theorems are no longer quoted in the proofs but are only referred to by paragraph numbers; while with increasing frequency the student is left to his own devices in supplying the reasons and even in filling in the logical steps of the argument. For convenience of reference the axioms and theorems of plane geometry which are used in the Solid Geometry are collected in the Introduction. In order to put the essential principles of solid geometry, together with a reasonable number of applications, within limited bounds (156 pages), certain topics have been placed in an Appendix. This was done in order to provide a minimum course in convenient form for class use and not because these topics, Similarity of Solids and Applications of Projection, are regarded as of minor importance. In fact, some of the examples under these topics are among the most interesting and concrete in the text. For example, see pages 180–183, 187–188, 194– 195. The exercises in the main body of the text are carefully graded as to difficulty and are not too numerous to be easily performed. The concepts of three-dimensional space are made clear and vivid by many simple illustrations and questions under the suggestive headings “Sight PREFACE Work.” This plan of giving many and varied simple exercises, so effec- tive in the Plane Geometry, is still more valuable in the Solid Geometry where the visualizing of space relations is difficult for many pupils. The treatment of incommensurables throughout the body of this text, both Plane and Solid, is believed to be sane and sensible. In each case, a frank assumption is made as to the existence of the concept in question (length of a curve, area of a surface, volume of a solid) and of its realization for all practical purposes by the approximation process. Then, for theoretical completeness, rigorous proofs of these theorems are given in Appendix III, where the theory of limits is presented in far simpler terminology than is found in current text-books and in such a way as to leave nothing to be unlearned or compromised in later mathematical work. Acknowledgment is due to Professor David Eugene Smith for the use of portraits from his collection of portraits of famous mathematicians. H. E. SLAUGHT N. J. LENNES Chicago and Missoula, May, 1919. CONTENTS INTRODUCTION 1 Space Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Axioms and Theorems from Plane Geometry . . . . . . . . . . 5 BOOK I. Properties of the Plane 10 Perpendicular Planes and Lines . . . . . . . . . . . . . . . . . 11 Parallel Planes and Lines . . . . . . . . . . . . . . . . . . . . . 21 Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Constructions of Planes and Lines . . . . . . . . . . . . . . . . 37 Polyhedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . 42 BOOK II. Regular Polyhedrons 53 Construction of Regular Polyhedrons . . . . . . . . . . . . . . 56 BOOK III. Prisms and Cylinders 58 Properties of Prisms . . . . . . . . . . . . . . . . . . . . . . . 59 Properties of Cylinders . . . . . . . . . . . . . . . . . . . . . . 75 BOOK IV. Pyramids and Cones 85 Properties of Pyramids . . . . . . . . . . . . . . . . . . . . . . 86 Properties of Cones . . . . . . . . . . . . . . . . . . . . . . . . 98 BOOK V. The Sphere 113 Spherical Angles and Triangles . . . . . . . . . . . . . . . . . 125 Area of the Sphere . . . . . . . . . . . . . . . . . . . . . . . . 143 Volume of the Sphere . . . . . . . . . . . . . . . . . . . . . . . 150 APPENDIX TO SOLID GEOMETRY I. Similar Solids . . . . . . . . . . . . . . . . . . . . . . . . . 168 II. Applications of Projection . . . . . . . . . . . . . . . . . . 183 III. Theory of Limits . . . . . . . . . . . . . . . . . . . . . . . . 196 INDEX 217 PORTRAITS AND BIOGRAPHICAL SKETCHES Cavalieri . . . . . . . . . . . . . . . . . . . . . . . . . Frontispiece Thales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Archimedes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Legendre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 SOLID GEOMETRY [...]... that the lower and right-hand sides are nearer the observer than the other edges Hence, the plane represented does not lie in the plane of the paper, but the lower part of it stands out toward the observer The figure ABCD represents a triangular pyramid The corner marked B is nearest the observer and this is indicated by the heavy lines The triangle ACD lies behind the pyramid and is thus farther from the. .. equal if the hypotenuse and a side of one are equal respectively to the hypotenuse and a side of the other 41 Two right triangles are equal if a side and an acute angle of one are equal respectively to the corresponding side and acute angle of the other 42 Two right triangles are equal if the hypotenuse and an acute angle of one are equal respectively to the hypotenuse and an acute angle of the other 43... area of a parallelogram is equal to the product of its base and altitude 59 Two parallelograms have equal areas if they have equal bases and equal altitudes 60 The area of a triangle is equal to one half the product of its base and altitude 61 If a is a side of a triangle and h the altitude on it and b another side and k the altitude on it, then ah = bk 62 The area of a trapezoid is equal to one half the. .. of the segment AB 5 Find the locus of all points equidistant from two given points A and B, and also equidistant from two points C and D Discuss 6 State and prove a theorem of solid geometry corresponding to the theorem of plane geometry given in § 38 7 If in the figure P D ⊥ plane M , and DC ⊥ AB, a line of the plane M , prove that P C ⊥ AB Suggestion Lay off CA = CB, and compare triangles 8 If in the. .. outside of each of two non-parallel lines there is one and only one plane parallel to both of these lines 100 Theorem XIII PROPERTIES OF THE PLANE 25 Given a point P outside of the non-parallel lines l1 and l2 To prove that there is one and only one plane M through P parallel to l1 and l2 Proof : Through P pass l3 l1 and l4 l2 Then the plane M of l3 and l4 is parallel to l1 and l2 In any other plane... in plane or in solid geometry? 7 Find the locus of all points equidistant from two given parallel planes Is this a problem in plane or in solid geometry? 8 The side walls of your schoolroom meet each other in four vertical lines Are any two of these parallel? Are any three of them parallel? Do any three of them lie in the same plane? 9 The side walls of your schoolroom meet the floor and the ceiling in... bisects the other side also 2 Show that a plane containing one only of two parallel lines is parallel to the other 3 If in two intersecting planes a line of one is parallel to a line of the other, then each of these lines is parallel to the line of intersection of the planes 4 Show that three lines which do not meet in one point must all lie in the same plane if each intersects the other two 26 SOLID GEOMETRY: ... angles 31 The sum of all consecutive angles about a point and on one side of a straight line is two right angles 32 If two adjacent angles are supplementary, their exterior sides lie in the same straight line 33 If in two triangles two sides of one are equal respectively to two sides of the other, but the third side of the first is greater than the third side of the second, then the included angle of the. .. parallel lines l1 and l2 To prove that in each case a plane is determined Proof : (1) Let A and B be two points on l Then one and only one plane M can be passed through l and P because one and only one plane can be passed through A, B, and P Ax 2, § 66 (2) Let A be the intersection point of l1 and l2 , and B and C any other points, one on l1 and the other on l2 Then A, B, and C determine the plane N in... to the plane of these lines 76 Theorem III Given a line l perpendicular to each of the lines l1 and l2 at the point P To prove that l is perpendicular to the plane of l1 and l2 Proof : Let M be the plane of l1 and l2 , and let l3 be any line in M through P Draw a line meeting l1 , l2 , and l3 in the points B, C, and D respectively Let E and F be points on l, on opposite sides of P , and such that . The Project Gutenberg EBook of Solid Geometry with Problems and Applications (Revised edition), by H. E. Slaught and N. J. Lennes This eBook is for the use of anyone anywhere at no cost and with almost. or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www .gutenberg. org Title: Solid Geometry with Problems and Applications (Revised edition) Author:. emphasized in the Solid Geometry. Owing to the greater maturity of the pupils it has been possible to make the logical structure of the Solid Geometry more prominent than in the Plane Geometry. The axioms
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